Finite Group Databasebeta

← #32981963

Subgroups — #32981963

Order 18688.

580 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618972C211yesmaximal, characteristic, complemented
2C4×#14556506C211yesmaximal, characteristic, complemented
3#22608447C212yesmaximal, complemented
4#22607783C212yesmaximal, complemented
5C73×#2635C211yesmaximal, characteristic
6#17988665—24no—
7#17988640—21no—
8C2×#14556506—21no—
9#17988551—21no—
10#17988551C2212yes—
11C4×C22⋊Dic73C2212yes—
12#17988354—21no—
13#17988321—24no—
14C2×#121×C73C2211yescharacteristic
15C2×C4×D4×C73C2211yescharacteristic
16#400×C73C411yescharacteristic, complemented
17#399×C73C411yescharacteristic, complemented
18#394×C73—21no—
19#394×C73C411yescharacteristic, complemented
20#392×C73—21no—
21#392×C73C411yescharacteristic, complemented
22#388×C73—21no—
23#386×C73—21no—
24#381×C73C2211yescharacteristic
25C2×#11924012—22no—
26C2×#11924012—22no—
27C2×C22⋊Dic73—22no—
28C2×C4×Dic73—22no—
29C4⋊#10596831—22no—
30#14556506—24no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33C22⋊#10596831D414yescomplemented
34C4×#10596831—22no—
35#14556498—22no—
36C22×D4×C73—21no—
37C22×D4×C73—21no—
38C22×D4×C73C2×C411yescharacteristic
39C22×D4×C73D411yescharacteristic
40C23×C4×C73D411yescharacteristic
41C23×C4×C73C2311yescharacteristic
42#121×C73—22no—
43#121×C73—22no—
44C4×D4×C73—22no—
45C4×D4×C73—22no—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C4⋊C4×C73C2×C411yescharacteristic
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73C2×C411yescharacteristic
56C2×C22⋊C4×C73D411yescharacteristic
57C2×C42×C73D411yescharacteristic
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61#95×C73C2×C411yescharacteristic
62#95×C73C2×C411yescharacteristic
63C22×Dic73—42no—
64#11924012—82no—
65#11924012—82no—
66C2×#10596831—24no—
67C22⋊Dic73—24no—
68C22⋊Dic73—42no—
69C4×Dic73—82no—
70C24×C73—21no—
71C24×C73C22⋊C411yescharacteristic
72C24×C73C2×D411yescharacteristic
73C2×D4×C73—22no—
74C2×D4×C73—22no—
75C2×D4×C73—22no—
76C2×D4×C73—41no—
77C2×D4×C73—42no—
78C2×D4×C73—42no—
79C2×D4×C73—81no—
80C2×D4×C73—81no—
81C2×D4×C73—81no—
82C2×D4×C73—81no—
83C2×D4×C73—81no—
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—81no—
90C22×C4×C73—81no—
91C22×C4×C73C22⋊C411yescharacteristic
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22⋊C411yescharacteristic
94C22×C4×C73C22×C411yescharacteristic
95C22×C4×C73C2×D412yes—
96C22×C4×C73C2×D411yescharacteristic
97C22×C4×C73#4112yes—
98C4⋊C4×C73—22no—
99C4⋊C4×C73—22no—
100C4⋊C4×C73—81no—
101C22⋊C4×C73—22no—
102C22⋊C4×C73—22no—
103C22⋊C4×C73—42no—
104C22⋊C4×C73—41no—
105C22⋊C4×C73—41no—
106C22⋊C4×C73—81no—
107C22⋊C4×C73—81no—
108C22⋊C4×C73—81no—
109C22⋊C4×C73—81no—
110C42×C73—21no—
111C42×C73—21no—
112C23×C73—21no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73C2×C22⋊C411yescharacteristic
129C23×C73#12112yes—
130D4×C73—44no—
131D4×C73—82no—
132D4×C73—84no—
133D4×C73—84no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—42no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—81no—
155C2×C4×C73#10412yes—
156C2×C4×C73C4×D412yes—
157C2×C4×C73#12011yescharacteristic
158C2×C4×C73#12311yescharacteristic
159C2×Dic73—44no—
160C2×Dic73—82no—
161C2×Dic73—82no—
162#10596831—44no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—22no—
166C22×C73—22no—
167C22×C73—22no—
168C22×C73—42no—
169C22×C73—41no—
170C22×C73—41no—
171C22×C73—41no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73—81no—
175C22×C73—81no—
176C22×C73—81no—
177C22×C73—81no—
178C22×C73—81no—
179C22×C73—81no—
180C22×C73—81no—
181C22×C73—81no—
182C22×C73#38611yescharacteristic
183C22×C73C2×#10411yescharacteristic
184C22×C73#42111yescharacteristic
185C4×C73—22no—
186C4×C73—21no—
187C4×C73—21no—
188C4×C73—22no—
189C4×C73—41no—
190C4×C73—41no—
191C4×C73—81no—
192C4×C73—81no—
193C4×C73—81no—
194Dic73—84no—
195#9186—731nomaximal, Sylow
196C2×C73—21no—
197C2×C73—21no—
198C2×C73—42no—
199C2×C73—81no—
200C2×C73—161no—
201C2×C73#187111yescharacteristic
202C2×C73#191011yescharacteristic
203C2×C73#191311yescharacteristic
204#2975—731no—
205#2635D7311yescharacteristic
206C4×#99—731no—
207#1507—732no—
208#1284—732no—
209C73#918611yescharacteristic, complemented, Sylow
210C2×#121C2×D7311yescharacteristic
211C2×C4×D4C2×D7311yescharacteristic
212#436—1464no—
213#431—1461no—
214C2×#99—1461no—
215#407—732no—
216#407—1461no—
217#400Dic7311yescharacteristic, complemented
218#399Dic7311yescharacteristic, complemented
219#394—21no—
220#394Dic7311yescharacteristic, complemented
221#392—21no—
222#392Dic7311yescharacteristic, complemented
223#388—21no—
224#386—21no—
225#381C2×D7311yescharacteristic
226C4×C22⋊C4—732no—
227#342—1461no—
228#327—1464no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2×Dic7311yescharacteristic
232C22×D4C22⋊D7311yescharacteristic
233C23×C4C22⋊D7311yescharacteristic
234C23×C4C22×D7311yescharacteristic
235C2×M16—1462no—
236C2×M16—1462no—
237#121—22no—
238#121—22no—
239C4×D4—22no—
240C4×D4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×Dic7311yescharacteristic
243C2×C4⋊C4C2×Dic7311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—1462no—
251C2×C22⋊C4C2×Dic7311yescharacteristic
252C2×C22⋊C4C22⋊D7311yescharacteristic
253C2×C42—1462no—
254C2×C42C22⋊D7311yescharacteristic
255C4⋊C8—1462no—
256#99—1464no—
257C22⋊C8—734no—
258C22⋊C8—1462no—
259C22⋊C8—1462no—
260C4×C8—1462no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—1462no—
265#95C2×Dic7311yescharacteristic
266#95C2×Dic7311yescharacteristic
267C24—21no—
268C24C22⋊Dic7311yescharacteristic
269C24C2×C22⋊D7311yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—81no—
280C2×D4—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—2922no—
289C22×C4C22⋊Dic7311yescharacteristic
290C22×C4C22⋊Dic7311yescharacteristic
291C22×C4C22⋊Dic7311yescharacteristic
292C22×C4C22×Dic7311yescharacteristic
293C22×C4D4×D7312yes—
294C22×C4C2×C22⋊D7311yescharacteristic
295C22×C4#1192402712yes—
296M16—5842no—
297M16—5842no—
298C2×C8—1464no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—81no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—42no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C22⋊C4—2922no—
312C22⋊C4—1464no—
313C42—21no—
314C42—21no—
315C42—5842no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23C2×C22⋊Dic7311yescharacteristic
333C23#1455656312yes—
334D4—44no—
335D4—82no—
336D4—84no—
337D4—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—2924no—
360C2×C4—5842no—
361C2×C4—5842no—
362C2×C4#1455651812yes—
363C2×C4D4×Dic7312yes—
364C2×C4#1455655911yescharacteristic
365C2×C4#1455657211yescharacteristic
366C8—2924no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1798846711yescharacteristic
387C22C2×#1455651811yescharacteristic
388C22#1798860411yescharacteristic
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—5844no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—81no—
403C2—161no—
404C2#2261023411yescharacteristic
405C2#2261045911yescharacteristic
406C2#2261047411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
Fitting, prime radicalC73×#2635C2
socleC22×C73#386
cosocleC23×C4×C73C23
hypercenter#2635D73
hypocenterC73#9186
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2635

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22618972C2semidirect
2. C4×#14556506C2semidirect
3. #22608447C2semidirect
4. #22607783C2semidirect
16. #400×C73C4semidirect
17. #399×C73C4semidirect
19. #394×C73C4semidirect
21. #392×C73C4semidirect
33. C22⋊#10596831D4semidirect
209. C73#9186semidirect
217. #400Dic73semidirect
218. #399Dic73semidirect
220. #394Dic73semidirect
222. #392Dic73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226189722
C4×#145565062
#226084472
#226077832
C73×#26352
#918673